LetMbe a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1, 2, . . . , n, is defined as the k-th ele-
mentary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a con-
formal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption
that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds
and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under
the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.